Recent Shell-Model Investigation and Its Possible Role in Nuclear Structure Data Study

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EPJ Web of Conferences 239, 04002 (2020) https://doi.org/10.1051/epjconf/202023904002 ND2019

Recent shell-model investigation and its possible role in nuclear structure data study

1, 1 1 1 1 Cenxi Yuan ∗, Yulin Ge , Menglan Liu , Guangshang Chen , and Boshuai Cai 1Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai, 519082, Guangdong, China

Abstract. Up to now, the nuclear shell model is rarely used in the nuclear data study because of several reasons. First, medium and heavy mass nuclei far from the shell-model cores, normally doubly magic nuclei, require a huge amount of calculation resource even in a limited shell-model space. Second, large deformation is diﬃcult to be described in the limited model space, which is based on spherical symmetry. Third, high precision evaluation of nuclear structure data challenges the ability of the shell model. Even so, it is worth starting preliminary nuclear data investigations based on the shell model. With the present computational ability, it is possible to investigate 1000 or more nuclei in the framework of the shell model, which should be helpful for nuclear data study. In the present work, some recent shell-model investigations are brieﬂy introduced. Based on these works, a simple nuclear force is suggested to be used in the systematic nuclear structure data study. The south-west region of 132Sn is taken as an example to show the ability of such a simple nuclear force.

1 Introduction than 20 years ago. It is worth performing systematic com- parisons between the observed nuclear structure data and Nuclear data is one of the key connections between nu- the shell-model results. If the structure properties men- clear physics and its application in many fields. For ex- tioned above are well reproduced in the framework of the ample, both the investigations on the nucleosynthesis in shell model, the further predictions on the unknown prop- the universe and the chain reactions in the nuclear power erties should be more reliable. Besides structure proper- plant require high precision nuclear data, such as nuclear ties, SM can provide spectroscopic factors, which are im- structure data including masses, half-lives, levels, neutron portant to cross section study [6, 7]. separation energies, β decay properties, and ratios of de- In the present work, some of our recent shell-model layed neutrons. works will be firstly reviewed, including both the theoreti- However, many nuclei involved in the nucleosynthesis cal analysis and comparison with observations. Secondly, and fission products in the reactor are beyond the present the south-west region of 132Sn is taken as an example to experimental abilities. Therefore, many nuclear structure show the ability of a simple nuclear force on the descrip- data should be theoretically evaluated, such as the nuclear tion of observed levels. masses evaluated by the finite-range droplet model [1], the Hartree-Fock-Bogoliubov model [2], and the Weizsäcker- Skyrme mass model [3]. The nuclear shell model (SM) 2 Recent Shell-model Study in Light, with configuration mixing is one of the most important Medium, and Heavy Mass Nuclei models to understand the underlying physics of atomic nu- A Hamiltonian, YSOX, was suggested for psd region [8]. clei [4, 5]. In SM, the eigenvalues (binding energies) and The neutron drip line of the B, C, N, and O isotopes are re- the eigenstates (wave functions) of both the ground and produced by YSOX but not by the previous Hamiltonians excited states are obtained simultaneously through the di- WBT and WBP [9]. 22C was predicted to be rather weakly β agonalization process. The electromagnetic properties, bound through YSOX and confirmed by the later experi- decays, and many other properties can be further calcu- ment [11] and AME2012 [12]. The quadrupole deforma- lated with the wave functions. tion of 16C is well described by YSOX [10]. The radius of SM is used to understand the nuclear structure prop- two neutron halo nucleus, 22C, was investigated based on erties of the extreme neutron- and proton-rich nuclei ob- the YSOX configuration and the neutron-neutron interac- served in recent decades, but rarely used in the nuclear tion [13]. In YSOX, the cross-shell interaction, especially data study. One obvious reason is that the calculation cost its off-diagonal part, was reconsidered. It was shown that of the nuclear shell model is vast, if the number of valence the levels, electromagnetic transitions, and B(GT) values nucleons is large. Up to the present calculation limitation, 14 11 of C can be well described only with a 4 ω model space around 10 dimensions, it is possible to calculate 1000 or and a weaker off-diagonal interaction [14]. The recent ob- more nuclei through the shell model, which is much more served neutron cross-shell properties of 11,12Be are well re- ∗e-mail: [email protected] produced by YSOX [15–17]. It is expected to investigate

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the island of inversion in sdp f region, with the northern 208Pb, 207Tl, and 206Hg are well reproduced, which indi- boundary at 34Si [18], through a similar method. cates the proper description of the N=126 gap. In sd region, phenomenological Hamiltonians, USD [19], USDA and USDB [20], are very successful, while realistic Hamiltonians can be obtained through ab initio method and Vlow k approach [21]. All three Hamil- − tonians in USD family are isospin symmetric without any mirror energy differences (MED), which are generally rather small. But some MED in nuclei around A= 20 are very large because of the weakly bound effect due to the protons in 1s1/2 orbit [22, 23]. With the inclusion of such effect in both single-particle energies and two-body matrix elements, the modified USD family can well describe MED in nuclei around A= 20 [23]. The recently measured β decay properties are well described through the modified Hamiltonian for 22Si [24], 27S [25, 26], and 26P [27]. Figure 1. Root mean square of nearly 200 levels of isotopes and 132 208 In the medium mass region, the nuclei around 132Sn are isotones of Sn and Sn as the function of strength of C11, LS1, and T1 terms especially important in nuclear data study due to their im- portance in both nucleosynthesis through r-process in the universe and the reactions in the reactor. For example, fission product yield has a peak in nuclei around A=140, of which the nuclear data contribute to the calculation of the present nuclear power plant [28]. The structure of 120,122I, 140Te, and 140I were investigated recently, showing proton- neutron bands and isomers in 120,122I [29, 30], the vibrator character of 140Te [31], and the suppression of the B(GT) from 140Te to 140I [32]. In addition, the monopole effect is shown to be important for the description of core excitation and B(GT) in A = 130 nuclei [33]. A Hamiltonian was suggested for the south-east region of 132Sn [34]. Based on the nice description of the observed neutron separation energies and levels, 8 isomers were predicted [34]. A later publication reported the first observation of spectroscopic property in this region, + 132 21 state in Cd with excitation energy 618(8) keV [35], while our Hamiltonian gives 710 keV. Recently, a new iso- mer was found in 134In [36], which showed similar decay energy and transition strength to our prediction. In the heavy mass region, the nuclear data of actinides are important for the transmutation of minor actinides in spent fuel [37] and the power distribution inside fuel rods of a thermal reactor [38–40]. It is impossible for SM to calculate all nuclei in this region at present, but the properties of nuclei close to 208Pb can be well reproduced. For example, the ground state spin parity of 223Np was observed with two possibilities, 9/2− and 7/2−, while SM 218 Figure 2. Upper panel: Root mean square of nearly 200 levels result supports the 9/2− [41]. The isomeric state of Pa of isotopes and isotones of 132Sn and 208Sn as the function of is recently observed [42], but the spins and parities of the strength of C10 term. Lower panel: Distribution of the strength ground and isomeric states can not be experimentally de- of C10 term. termined with the present statistics. Different to the systematic trends of N= 127 isotones from 210Bi to 216Ac, which have 1− ground states and high spin (8− or 9−) iso- 218 mers, the ground and isomeric states of Pa are suggested 3 Applied VMU to Nuclear Structure Data to be 8− and 1− states, respectively, based on SM results Study [42]. A new Hamiltonian for the south region of 208Pb is under preparation, which described 45 observed binding Monopole based universal interaction, VMU, is suggested energies within a root mean square (RMS) 0.09 MeV [43]. through the monopole properties of central and tensor 209 In addition, the neutron core excitation states of Pb, force [44]. VMU plus spin-orbit interaction from M3Y

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126 130 208 207 206 Observed and calculated levels of − Sn and the island of inversion in sdp f region, with the northern Pb, Tl, and Hg are well reproduced, which indi- (VMU+LS) is used as the cross-shell interaction in many Table 1. 127 130 boundary at 34Si [18], through a similar method. cates the proper description of the N=126 gap. regions, such as psd [8] and 132Sn [34]. It is worth know- − In. Observed data are taken from NNDC [49]. ing the performance of the nuclear structure data purely In sd region, phenomenological Hamiltonians, nucleus Jπ Expt. V +LS jj45pnm jj45pna from V +LS. V +LS includes 8 terms, central force MU USD [19], USDA and USDB [20], are very successful, MU MU 126Sn 0+ 0.00 0.00 0.00 0.00 with T=1, S=0 (C10), T=1, S=1 (C11), T=0, S=0 (C00), while realistic Hamiltonians can be obtained through ab 2+ 1.14 1.31 1.15 1.27 and T=0, S=1 (C01), spin-orbit force with T=1 (LS1) and initio method and Vlow k approach [21]. All three Hamil- 4+ 2.05 1.92 2.06 2.28 − T=0 (LS0), and tensor force with T=1 (T1) and T=0 (T0). tonians in USD family are isospin symmetric without any 5- 2.16 2.13 2.25 2.49 If semi-magic nuclei are considered, only four T=1 terms mirror energy differences (MED), which are generally 7- 2.22 2.12 2.32 2.57 contribute to the nuclear structure properties. rather small. But some MED in nuclei around A= 20 6- 2.48 2.19 2.49 2.76 Considering nearly 200 levels in isotopes and isotones are very large because of the weakly bound effect due to (8+) 2.49 2.20 2.51 2.78 of 132Sn and 208Pb, it is found that a stronger strength the protons in 1s1/2 orbit [22, 23]. With the inclusion of (10+) 2.56 2.26 2.54 2.81 of C10 is needed for proton-proton interaction compar- such effect in both single-particle energies and two-body 127Sn 11/2- 0.00 0.00 0.00 0.00 ing with that of neutron-neutron interaction [45]. Levels matrix elements, the modified USD family can well 3/2+ 0.01 0.05 0.03 0.04 of 123,125Ag are well reproduced through pure V +LS describe MED in nuclei around A= 20 [23]. The recently MU (1/2)+ 0.26 0.19 0.18 0.19 with the strength for C10 term of proton-proton interac- measured β decay properties are well described through (9/2)- 0.65 0.93 0.76 0.85 22 27 tion 10% stronger than that of neutron-neutron interaction, the modified Hamiltonian for Si [24], S [25, 26], and (5/2+) 0.81 1.04 0.91 1.01 26 which are 115% and 105% of the original strength, respec- P [27]. Figure 1. Root mean square of nearly 200 levels of isotopes and (7/2-) 0.96 1.14 0.88 0.96 132 208 tively [46]. The type II shell evolution is suggested in each In the medium mass region, the nuclei around 132Sn are isotones of Sn and Sn as the function of strength of C11, (7/2+) 1.05 1.24 1.29 1.43 LS1, and T1 terms neutron deficient In isotope from 101In to 109In based on especially important in nuclear data study due to their im- (15/2-) 1.09 1.24 1.04 1.15 SM calculations through such nuclear force with a little portance in both nucleosynthesis through r-process in the 128Sn 0+ 0.00 0.00 0.00 0.00 modifications [47]. In the present work, we further inves- universe and the reactions in the reactor. For example, fis- (2)+ 1.17 1.31 1.23 1.36 tigate the contribution of C11, LS1, and T1 terms. As seen sion product yield has a peak in nuclei around A=140, of (4+) 2.00 1.87 2.03 2.24 in figure 1, the strengths of C11, LS1, and T1 change from which the nuclear data contribute to the calculation of the (7-) 2.09 1.99 2.14 2.37 80% to 120% of their original values, but the RMS of lev- present nuclear power plant [28]. The structure of 120,122I, (5-) 2.12 2.03 2.17 2.40 els are rarely affected by these three terms. 140Te, and 140I were investigated recently, showing proton- 129Sn 3/2+ 0.00 0.00 0.00 0.00 More detailed investigations on C10 term are also per- neutron bands and isomers in 120,122I [29, 30], the vibrator 11/2- 0.04 0.01 0.05 0.05 formed, as seen in the upper panel of figure 2. If the iso- character of 140Te [31], and the suppression of the B(GT) (1/2)+ 0.32 0.23 0.31 0.32 topes and isotones of 132Sn and 208Pb are considered to- from 140Te to 140I [32]. In addition, the monopole effect is (9/2-) 0.76 1.00 1.14 1.24 gether, around 107% of original C10 strength gives the shown to be important for the description of core excita- (5/2+) 0.77 0.95 0.96 1.06 minimum of RMS of levels at around 0.2 MeV. But if tion and B(GT) in A = 130 nuclei [33]. (7/2-) 1.04 1.16 1.04 1.13 isotopes and isotones of 132Sn and 208Pb are separately (7/2+) 1.05 1.16 1.21 1.34 A Hamiltonian was suggested for the south-east reconsidered, around 103% (113%) of original C10 strength 132 (15/2-) 1.17 1.27 1.24 1.35 gion of Sn [34]. Based on the nice description of the gives the minimum of RMS of levels at around 0.15 (0.15) 130Sn 0+ 0.00 0.00 0.00 0.00 observed neutron separation energies and levels, 8 iso- MeV for isotopes contributing by neutron-neutron interac- (2+) 1.22 1.33 1.42 1.56 mers were predicted [34]. A later publication reported the tion (isotones contributing by proton-proton interaction). (7-) 1.95 1.86 1.92 2.13 first observation of spectroscopic property in this region, With the bootstrap statistical method, the distribution of + 132 (4+) 2.00 1.85 2.13 2.35 21 state in Cd with excitation energy 618(8) keV [35], the strength is obtained, as seen in the lower panel of (5-) 2.08 1.96 2.08 2.30 while our Hamiltonian gives 710 keV. Recently, a new iso- figure 2. It is clearly shown that the distribution of the 134 (4-) 2.21 1.85 2.17 2.41 mer was found in In [36], which showed similar decay strength for C10 term of proton-proton interaction has (6+) 2.26 2.01 2.34 2.58 energy and transition strength to our prediction. no overlap with that of neutron-neutron interaction. The (8+) 2.34 2.08 2.42 2.67 In the heavy mass region, the nuclear data of actinides strength of proton-proton interaction is statistically differ- (10+) 2.43 2.16 2.48 2.73 are important for the transmutation of minor actinides in ent from that of neutron-neutron interaction. 127In (9/2+) 0.00 0.00 0.00 0.00 spent fuel [37] and the power distribution inside fuel rods Table 1 and 2 present the SM results from V +LS MU (1/2-) 0.41 0.05 0.04 0.03 of a thermal reactor [38–40]. It is impossible for SM to with different strengths for proton and neutron parts. Ob- (3/2-) 0.93 0.62 0.37 0.38 calculate all nuclei in this region at present, but the prop- served data and results from jj45pna and jj45pnm are also (21/2-) 1.86 1.88 1.99 2.22 erties of nuclei close to 208Pb can be well reproduced. For presented. Nuclei in the south-west region of 132Sn with 128In (3)+ 0.00 0.00 0.00 0.00 example, the ground state spin parity of 223Np was ob- observed levels and A 126 are considered as exam- ≥ (1-) 0.25 0.06 0.32 0.29 served with two possibilities, 9/2− and 7/2−, while SM ples. jj45pna is from G matrix calculation and included 218 Figure 2. Upper panel: Root mean square of nearly 200 levels (8-) 0.34 0.65 1.05 1.06 result supports the 9/2 [41]. The isomeric state of Pa 132 208 in OXBASH package [48]. jj45pnm is modified version − of isotopes and isotones of Sn and Sn as the function of 1+ 1.17 0.97 0.65 0.66 is recently observed [42], but the spins and parities of the of jj45pna with 77% (92%) of proton-proton (neutron- strength of C10 term. Lower panel: Distribution of the strength 129In (9/2+) 0.00 0.00 0.00 0.00 ground and isomeric states can not be experimentally de- neutron) interaction. Such modifications give a better de- of C10 term. (1/2-) 0.46 0.21 0.16 0.15 termined with the present statistics. Different to the sys- scription on levels of nuclei around 132Sn. Single-particle (11/2+) 1.00 1.07 1.33 1.46 tematic trends of N= 127 isotones from 210Bi to 216Ac, energies of the three Hamiltonians are taken to be the same (3/2-) 1.09 0.87 0.65 0.67 which have 1 ground states and high spin (8 or 9 ) iso- as the observed levels of 131Sn and 131In. − − − (13/2+) 1.35 1.28 1.34 1.45 mers, the ground and isomeric states of 218Pa are suggested Comparing with 88 levels in 15 nuclei, the RMS of 3 Applied VMU to Nuclear Structure Data (5/2+) 1.42 1.57 1.49 1.58 to be 8 and 1 states, respectively, based on SM results levels are 0.17, 0.27, and 0.47 for V +LS, jj45pnm, and − − Study MU 130In 1(-) 0.00 0.00 0.00 0.00 [42]. A new Hamiltonian for the south region of 208Pb is jj45pna, respectively, as shown in table 3. Simple nuclear (10-) 0.05 0.35 0.28 0.28 under preparation, which described 45 observed binding Monopole based universal interaction, V , is suggested force V +LS gives a very nice description of the ob- MU MU (3+) 0.39 0.40 0.46 0.46 energies within a root mean square (RMS) 0.09 MeV [43]. through the monopole properties of central and tensor served levels. Considering the RMS of individual nucleus, (5+) 0.40 0.59 0.56 0.57 In addition, the neutron core excitation states of 209Pb, force [44]. V plus spin-orbit interaction from M3Y V +LS provides rather stable descriptions without very MU MU 1+ 2.12 1.79 1.38 1.44

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126,128 130 126,128 Table 2. The same as table 1 but for − Cd and Pd. large RMS in each nucleus. But for jj45pnm and jj45pna, nuclei with strong proton and neutron configuration mix- nucleus Jπ Expt. V +LS jj45pnm jj45pna MU ing are poorly described, such as for 128In, 128Cd, and 126Cd 0+ 0.00 0.00 0.00 0.00 126Pd. Even though further and more systematic investi- (2+) 0.65 0.91 0.78 0.90 gation should be carefully performed, the present results (4+) 1.47 1.73 1.68 1.90 show the possibility of applying V to nuclear struc- 128Cd 0+ 0.00 0.00 0.00 0.00 MU ture data study. The uncertainty of such a theoretical de- (2+) 0.65 0.90 0.95 1.13 scription also needs to be taken into account through sta- (4+) 1.43 1.65 1.91 2.21 tistical methods, such as the uncertainty decomposition (5-) 1.87 1.87 1.86 2.19 method [50] and the bootstrap statistical method [51]. The (7-) 2.11 1.97 2.20 2.44 uncertainties of the observed data are also not included in (6+) 2.20 2.18 2.75 3.14 the present fitting, which should be considered if they are (8+) 2.65 2.48 3.18 3.63 as large as the theoretical ones (around 0.1 MeV). (10+) 2.71 2.55 3.40 3.63 129Cd 11/2- 0.00 0.00 0.00 0.00 (13/2-) 1.18 1.22 1.44 1.86 4 Conclusion (15/2-) 1.59 1.60 1.82 2.20 (21/2+) 1.94 2.11 2.05 2.53 Based on some recent shell-model investigations, a sim- 130Cd 0+ 0.00 0.00 0.00 0.00 ple nuclear force, VMU+LS is suggested to be applied to (2+) 1.33 1.34 1.40 1.96 nuclear structure data study. As an example, levels of nu- (4+) 1.86 1.81 1.93 2.71 clei in the south-west region of 132Sn are investigated. It is (6+) 1.99 1.97 2.11 2.96 shown that VMU+LS can give very nice descriptions of the (8+) 2.13 2.08 2.22 3.10 observed levels, with 0.17 MeV RMS. It is worth to per- 126Pd 0+ 0.00 0.00 0.00 0.00 form systematic investigations on the medium and heavy (2+) 0.69 0.98 0.82 1.05 mass region to see the performance of VMU+LS on the (4+) 1.48 1.83 1.66 2.07 binding energies, levels, electromagnetic transitions, and (5-) 2.02 2.03 2.09 2.54 many other properties. (7-) 2.11 1.98 2.26 2.55 + (10 ) 2.41 2.61 3.73 3.80 Acknowledgments 128Pd 0+ 0.00 0.00 0.00 0.00 (2+) 1.31 1.41 1.46 2.03 The shell-model calculations are performed with code (4+) 1.82 1.90 2.01 2.83 KSHELL [52]. This work has been supported by the Na- (6+) 2.08 2.08 2.20 3.09 tional Natural Science Foundation of China under Grant (8+) 2.15 2.18 2.30 3.22 No. 11775316, the Tip-top Scientific and Technical Inno- vative Youth Talents of Guangdong special support pro- Table 3. Root mean square between observed data and gram under Grant No. 2016TQ03N575, the National Key calculations for each nucleus and all nuclei in tables 1 and 2. Research and Development Program of China under Grant

nucleus VMU+LS jj45pnm jj45pna No. 2018YFB1900405, the computational resources from 126Sn 0.20 0.05 0.26 SYSU and National Supercomputer Center in Guangzhou. 127Sn 0.17 0.11 0.17 128 Sn 0.10 0.04 0.22 References 129Sn 0.13 0.16 0.23 130Sn 0.21 0.09 0.27 [1] P. Möller and J.R. Nix, At. Data Nucl. Data Tables 59, 127In 0.24 0.34 0.38 185 (1995). 128 In 0.21 0.44 0.44 [2] S. Goriely, et al., Phys. Rev. C 88, 061302(R) (2013). 129 In 0.16 0.26 0.29 [3] N. Wang, et al., Phys. Lett. B 734, 215 (2014). 130 In 0.22 0.35 0.33 [4] B. A. Brown, Prog. Part. Nucl. Phys. 47, 517 (2001). 126Cd 0.21 0.14 0.29 [5] E. Caurier, et al., Rev. Mod. Phys. 77, 427 (2005). 128Cd 0.15 0.42 0.68 [6] Y.P. Xu, et al., Phys. Rev. C 98, 044622 (2018). 129Cd 0.09 0.18 0.54 [7] Y.P. Xu, et al., Phys. Lett. B , 308 (2019). 130Cd 0.04 0.08 0.77 790 126Pd 0.21 0.55 0.69 [8] C.X. Yuan, et al., Phys. Rev. C 85, 064324 (2012). 128Pd 0.06 0.14 0.86 [9] E. K. Warburton and B. A. Brown, Phys Rev. C 46, total 0.17 0.27 0.47 923 (1992). [10] Y. Jiang, et al., Phys. Rev. C 101, 024601 (2020). [11] L. Gaudefroy, et al., Phys. Rev. Lett. 109, 202503 (2012). [12] M. Wang, et al., Chin. Phys. C, 41(3), 030003 (2017). [13] T. Suzuki, et al., Phys. Lett. B 753, 199 (2016).

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126,128 130 126,128 Table 2. The same as table 1 but for − Cd and Pd. large RMS in each nucleus. But for jj45pnm and jj45pna, [14] C.X. Yuan, Chin. Phys. C, 41(10), 104102, (2017). [36] V.H. Phong, et al., Phys. Rev. C 100, 011302(R) nuclei with strong proton and neutron configuration mix- (2019). nucleus Jπ Expt. V +LS jj45pnm jj45pna [15] J. Chen, et al., Phys. Rev. C 100, 064314 (2019). MU ing are poorly described, such as for 128In, 128Cd, and 126Cd 0+ 0.00 0.00 0.00 0.00 [16] J. Chen, et al., Phys. Lett. B 781, 412 (2018). [37] S.L. Chen and C.X. Yuan, Ann. Nucl. Energy 127, 126Pd. Even though further and more systematic investi- (2+) 0.65 0.91 0.78 0.90 [17] J. Chen, et al., Phys. Rev. C 98, 014616 (2018). 204 (2019). gation should be carefully performed, the present results (4+) 1.47 1.73 1.68 1.90 [18] R. Han, et al., Phys. Lett. B 772, 529 (2017). [38] C.X. Yuan, et al., Sci. Techol. Nucl. 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This work has been supported by the Na- (6+) 2.08 2.08 2.20 3.09 tional Natural Science Foundation of China under Grant (8+) 2.15 2.18 2.30 3.22 No. 11775316, the Tip-top Scientific and Technical Inno- vative Youth Talents of Guangdong special support pro- Table 3. Root mean square between observed data and gram under Grant No. 2016TQ03N575, the National Key calculations for each nucleus and all nuclei in tables 1 and 2. Research and Development Program of China under Grant nucleus VMU+LS jj45pnm jj45pna No. 2018YFB1900405, the computational resources from 126Sn 0.20 0.05 0.26 SYSU and National Supercomputer Center in Guangzhou. 127Sn 0.17 0.11 0.17 128 Sn 0.10 0.04 0.22 References 129Sn 0.13 0.16 0.23 130Sn 0.21 0.09 0.27 [1] P. Möller and J.R. Nix, At. Data Nucl. Data Tables 59, 127In 0.24 0.34 0.38 185 (1995). 128 In 0.21 0.44 0.44 [2] S. Goriely, et al., Phys. Rev. C 88, 061302(R) (2013). 129 In 0.16 0.26 0.29 [3] N. Wang, et al., Phys. Lett. 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